Lesson 6.3

Vertical Asymptotes

Where the graph explodes to infinity. Understanding the invisible walls that functions cannot cross.

Introduction

Prerequisite Connection: In Lesson 6.2, we found "problem numbers" where the denominator was zero. In this lesson, we see what the graph actually DOES at those numbers.

Today's Increment: A Vertical Asymptote (VA) is a vertical line . As the graph gets closer and closer to this line, the y-values shoot up to or crash down to .

Visual: Think of it as a force field. The graph can get infinitely close, but it can never touch or cross a Vertical Asymptote.

Explanation of Key Concepts

How to Find Them

Step 1: Simplify FIRST!

You MUST factor and cancel common terms before finding asymptotes. (If they cancel, they are Holes, not VAs—see Lesson 6.5).

Step 2: Set Denominator = 0

Whatever factors are LEFT in the bottom will create Vertical Asymptotes.

Infinite Behavior

Odd Multiplicity (Example:

)

"Opposite Directions"

[Graph goes UP on one side, DOWN on other]

Like a Volcanic Eruption vs Waterfall

Even Multiplicity (Example:

)

"Same Direction"

[Both sides go UP or both go DOWN]

Like a Chimney or a Pit

Worked Examples

Level: Basic

Example 1: Finding Equations

Find the vertical asymptotes of .

Is it simplified? Yes. Nothing cancels.

Set Bottom = 0

Answer

Two Vertical Asymptotes:

(The y-axis)

Level: Intermediate

Example 2: Hidden Asymptotes

Find VAs for .

Factor Denom:

Set factors to zero:

Equations:

and

Level: Advanced

Example 3: Behavior at the Line

Describe the behavior near the asymptote for .

VA is at

.
The factor is squared (

), so Multiplicity = 2 (Even).

Interpretation: Since square numbers are always positive, the denominator is always positive (but tiny) near 3.

Conclusion

on BOTH sides of

Common Pitfalls

Saying "The Asymptote is 5":
It's a LINE, not a number. You MUST write "". Writing just "5" is wrong because that could mean .
Confusing Holes and Asymptotes:
If is in the top AND bottom, it cancels out. That's a Hole. It is NOT an asymptote. Only simplify first!

Real-World Application

Black Holes

In relativity, the gravitational force near a black hole is modeled by equations with vertical asymptotes at the "Event Horizon" (Schwarzschild radius).

As you approach , the time dilation approaches infinity. To an outside observer, time literally stops at the asymptote.